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I is defined by i^2 = -1

  1. Jul 5, 2008 #1
    can any one tell me that what is i([tex]\sqrt{-1}[/tex])
     
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  3. Jul 5, 2008 #2

    Astronuc

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  4. Jul 5, 2008 #3
    Re: i

    I can not understand the existance of this
     
  5. Jul 5, 2008 #4

    HallsofIvy

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    Re: i

    It is impossible to give an explanation that you will grasp if we do not know what mathematics you already know.
     
  6. Jul 5, 2008 #5

    nicksauce

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    Re: i

    What do you mean? It is defined to exist. All you have to do is to be able to grasp the concept of definition.
     
  7. Jul 5, 2008 #6
    Re: i

    pre unive math
     
  8. Jul 5, 2008 #7

    arildno

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    Re: i

    Well, then you know about 2-D coordinates, don't you?

    Instead of considering the quantities that lie on the real number line (i.e, the so-called "real" numbers!), let our basic "quantities" be points in the PLANE in stead.
    We will call these quantities "complex numbers".

    Now, each such "complex number" can be said to have two components, each of which is a standard real number, just like the coordinates for a point in the plane are given by, say, (x,y) where "x" is the "x-coordinate" and "y" the "y-coordinate"

    Now, we want to DO SOMETHING with our new complex numbers, in particular, we want to be able to "multiply" two such numbers together. (Note that we here introduce a new meaning to the word "multiply" , namely as an operation between two points in the plane (i.e, our complex numbers!), rather than an operation between two numbers on the number line!)


    Let two complex numbers have the shape (a,b) and (c,d) where a,b,c,d are ordinary real numbers.
    Furthermore, let us introduce the short-hand symbol "i" for the point (0,1), that is i=(0,1), and in addition, we let the symbol ""1"" be defined as: "1"=(1,0), and, also, "-1"=(-1,0).


    Now, we introduce a general multiplication rule:

    (a,b)*(c,d)=(ac-bd,ad+bc)

    Thus, let us multiply "i" with itself!

    i*i=(0,1)*(0,1)=(0*0-1*1,0*1+1*0)=(-1,0)="-1"

    That is, i*i equals the complex number we have called "-1"!


    Furthermore, we will now look at those numbers that lie on just the x-axis, say numbers of the form (a,0) and (c,0).

    What happens when we multiply two such numbers together?

    Then we get:
    (a,0)*(c,0)=(ac-0*0,a*0+0*a)=(ac,0)
    THat is, we get a number ON the x-axis whose coordinate value equals exactly the product a*c, and we can therefore IDENTIFY the x-axis with our ordinary number line!!!


    THe complex numbers constitute therefore a number PLANE, where we call the x-axis the "real axis", whereas the y-axis is called the "imaginary axis".

    The number "i" lies therefore ON the "imaginary axis", has unit length, so we call it the "imaginary unit".




    Thus, that i*i=-1 is something we are able to construct as long as we regard our NUMBERS as points in the plane, and in addition, have defined our "multiplication" in a smart manner as above.
     
  9. Jul 5, 2008 #8

    tiny-tim

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    Hi hadi amiri 4! :smile:

    nicksauce is right … i is defined as a square-root of -1.

    Mathematicians can define anything they like.

    i does not exist.

    Or, rather, it only exists in mathematicians' imaginations.

    Why does its existence worry you? :smile:
     
  10. Jul 5, 2008 #9

    symbolipoint

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    Re: i

    You already know that i is the imaginary unit, so you then know that i([tex]/sqrt{-1}[/tex]) is the same as i*i, which is -1.
     
  11. Jul 5, 2008 #10

    Redbelly98

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    Re: i

    It occured to me some time ago that this definition is ambiguous, since there are two solutions to that equation.

    While this ambiguity does not seem to matter in practice, I'm still surprised that it never seems to get discussed.
     
  12. Jul 5, 2008 #11
    Re: i

    i exists just as much as the number 1. i is simply the element (0, 1) of the field formed by R^2 over addition and multiplication of complex numbers. That i^2 = -1 is simply due to the definition of multiplication for complex numbers. Also, to be clear, 1 in C is not the same as 1 in R. They are different objects; one is an ordered pair while the other is atomic, let alone the fact that they come from different fields. For convenience, we abbreviate (x, 0) as x and (0, y) as y*i. The pair (0, 1) exists just as much as the pair (1, 0).
     
    Last edited: Jul 5, 2008
  13. Jul 5, 2008 #12

    Gokul43201

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    Re: i

    Nice! But I suspect a student's ability to appreciate this will depend strongly on whether or not they have been taught mathematics axiomatically.
     
  14. Jul 5, 2008 #13
    Re: i

    Remember, math is a relatively arbitrary field. Something can be made valid so long as it serves a purpose that does not hold too many heavy contradictions. The number 1 is a concept and so is i. The number 0 is a concept and so is i (it can even be more mysterious than i).
    Pre-university schools (and even universities can) have a crude way of teaching math...its very mechanical and rigid...it does not present any elegance and it pretends to put unnecessary boundaries between the several different categories. You could gain a better perspective from reading journals, etc... (how Gokul43201 said it).
     
    Last edited: Jul 5, 2008
  15. Jul 6, 2008 #14

    HallsofIvy

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    Re: i

    Oh, tiny-tim, what were you thinking? "i" only exists in mathematician's imaginations in exactly the same way other abstract things, like the numbers "e", or "1/2", or "1" do. I understand that but the person who wrote the original post is certain to misunderstand it.

    Oh, and as I am sure you have seen from my previous posts, I object to the bald statement "i is defined as a square root of -1". You did put "a" which a good thing but a definition has to specify it exactly. That does not say which of the two square roots of -1 is i.
     
  16. Jul 6, 2008 #15

    Dale

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    Re: i

    I hadn't thought of that, both +i and -i are solutions, but how can you say that one is the positive root and the other is the negative root? Hmmm, very interesting comment. I guess that is the ultimate source of conjugate symmetry.
     
  17. Jul 6, 2008 #16

    arildno

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    Agreed!
    It would be interesting to see what the orinal poster makes out of it.
     
  18. Jul 6, 2008 #17

    matt grime

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    Re: i

    It is discussed. In one loose sense it is the starting point of Galois theory. In fact some mathematicians make a point of only using the symbol [itex]\sqrt{-1}[/itex] to avoid having to make a choice of square root of -1.
     
  19. Jul 6, 2008 #18

    Redbelly98

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    Thanks Matt, I just don't hang around with mathematician's enough to know what is going on. My exposure to math has been from a physics education earlier in life, and (these days) tutoring high school math.

    Mark
     
  20. Jul 6, 2008 #19

    tiny-tim

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    The most basic definition of i

    Hi Werg22!
    hmm … this is obviously a usage of the word "simply" that I haven't come across before … :wink:

    Hi HallsofIvy! :smile:
    Sorry, but I think numbers like 1 2 3 … existed even in cavemen's imaginations.

    And when it came to dividing food, so did numbers like 1/2.

    e = 2.71828182889045etc is of course a much more recent invention of mathematicians … but is it an invention?

    If you defined it (using that decimal series) it to a layman, surely the layman would reply "You haven't defined anything new … you've only given a name to something which I see no use for! I wish you well with it! But it was already there before you gave it that name … you didn't define it, you only named it!"

    But i wasn't there before …

    :rofl: if so, where was it? :rofl:​

    In that sense, hadi amiri 4 is quite right to question its existence! :smile:
    erm … sorry … what posts? :confused:
    "i" can be defined in many ways.

    Here's one definition: add a symbol "i" to the field R, define ii = -1, and then enlarge to a field.

    Obviously, there are no other elements of the new field whose square is -1, except for -i.

    What's wrong with that?

    All other definitions (the Argand plane construction, the pairs construction, etc) are just models of this definition.

    And this definition … of i as a thing whose square is -1 … is the most basic. :smile:
     
  21. Jul 6, 2008 #20

    matt grime

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    Re: The most basic definition of i

    So you don't think they exist in any real sense any more nor less than a symbol that squares to give -1?

    Just because you can think of a 'real life' situation where things like 1,2,3 are useful, and describe some properties doesn't mean that these descriptors have a physical reality in this, or any (platonic) realm. My standard line is: the second I stub my toe on a '1', I'll believe it exists in a meaningful sense. The counter point is: when you stick your hand in the electric socket, it isn't the imaginary part of the current that kills you.
     
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